Six Maps with a Common Fixed Point Satisfying Weak Compatible and Commuting Mapping in complex valued metric space Using Rational Inequalities

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ISSN(Online): 2319-8753 ISSN (Print): 2347-6710

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(An ISO 3297: 2007 Certified Organization)

Vol. 5, Issue 6, June 2016

Six Maps with a Common Fixed Point

Satisfying Weak Compatible and Commuting

Mapping in complex valued metric space

Using Rational Inequalities

Kamal Kumar1, Nisha Sharma2,*, Rajeev Jha3, Ritu Sharma4, Sheetal Sharma5

Department of Mathematics, Pt. JLN Govt. College Faridabad, Sunrise University Alwar, India1

Department of Mathematics, Pt. JLN Govt. College Faridabad, Faridabad, Haryana, India 2

Department of Mathematics, Teerthankar Mahaveer University, Moradabad, U.P, India 3

Department of Mathematics, Pt. JLN Govt. College Faridabad, India 4

Assistant Professor, Department of computer Science and Engineering, Amity university sector-125, Noida, India5

*

Corresponding Author

ABSTRACT. Azam et al. (2011), introduce the notion of complex valued metric spaces and obtained common fixed point result for mappings in the context of complex valued metric spaces. In this paper we prove common fixed point theorems for six maps in complex valued metric space satisfying commuting and weakly compatible mappingwith different type of inequality,our result generalizes the result of Tiwari et al.[8]

KEYWORDS: commuting mapping, weakly compatible maps, common fixed points, complete complex valued metric space

MSC: 46T99, 47H10, 54H25, 54C30

I. INTRODUCTION

Banach contraction principle was the starting point for many researchers during last few decades in the field of nonlinear analysis. The concept of complex valued metric pace which is a generalization of the classical metric space was recently introduced by Azam, Fisher and khan[1].The study of metric space expressed the most common important role to many fields both in pure and applied science [3]. Abounding researchers extended the notion of a metric space such as vector valued metric space of Perov [2], a cone metric spaces of Huang and Zhang [6], a modular metric spaces of Chistyakov[10], etc. For the sake of completeness we recollect some basic definitions and fundamentals results on the topic in the literature. We mainly follow the notions and notations of Azam et al.[1].

Let ℂ bethesetofallcomplexnumbers,for z1,z2∈ ℂ, define a partial order ≾ on ℂ asfollows;

z1≾z2 iff Re(z1) ≤ Re(z2), Im(z1) ≤ Im(z2) it follows that

z1≾z2 ,if one of the following conditions is satisfied:

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ii. Re(z1)<Re(z2), Im(z1)=Im(z2) iii. Re(z1)<Re(z2), Im(z1)<Im(z2) iv. Re(z1)=Re(z2), Im(z1)=Im(z2)

In particular, we will write z1⪱ z2 if z1≠z2 and one the (i), (ii), (iii) conditions is satisfied and we will write z1≺z2. Note that

0≾ z1⪱z2⇒ |z1|<|z2|,

z1≾z2 , z2≺z3⇒ z1≺z3

Definition 1.2.[1]Let X be a nonempty set. A mapping d:X x X→ ℂ is called a complex valued metric on X if the following conditions are satisfied:

(1.1) 0 ≾ d(x,y) for all x,y∈X and d(x,y)=0⇔ x=y.

(1.2) d(x,y)=d(y,x) for all x,y∈X

(1.3) d(x,y) ≾ d(x,z)+d(z,y) for all x,y,z∈X.

In this case, we say that (X,d) is a complex valued metric space.

Definition 1.3.[3]Let ℂ be a complex valued metric space,

 We say that a sequence {xn} is said to be a Cauchy sequenceX, if for every c∈ ℂ, with 0≺c there is n0∈ ℕsuch that for all n>n0 such thatd(xn,xm)≺c.

 We say that a sequence {xn} converges to an element x∈X. If for every c∈ ℂ, with 0≺c ther exist an integer n0∈ ℕ such that for all n>n0 such that d(xn,x)≺c and we write xn→x.

 We say that (X,d) is complete if every Cauchy sequence in X converges to a point in X.

Lemma 1.4.[3]Any sequence {xn} in complex valued metric space (X,d), converges to x if and only if |d(xn,x)|→0 as n→ ∞

Lemma 1.5.[3]Any sequence {xn} in complex valued metric space (X,d) is a Cauchy sequence if and only if |d(xn,xn+m)|→0 as n→ ∞, where m∈ ℕ

Definition 1.6.[9] Let S and T be self-maps of metric space (X,d), then S,T are said to be weakly commuting if d(STx,TSx)≤d(Sx,Tx), for all x∈X

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II. MAIN RESULTS

In this section, we prove some common fixed point theorems for rational type contraction conditions. Our main result

runs as follows

Theorem 2.1.Let (X,d) be acomplete complex valued metric space and P,Q,R,S,T,U be self-maps of X satisfying the following condition

(2.1) TU(X) ⊆P(X) and RS(X) ⊆Q(X) and

(2.2) d(RSx,TUy) ≾

ad(Px, Qy)[ (₍ , ) ( , )]

, ) ( , ) + bd(TUy, RSx)

[ ( , ) ( , )] ( , ) ( , )

+cd(Qy, RSx) ₍ [ ( , )]

, ) ( , )+ ed(Qy, RSx)

[ ( , )] ( , ) ( , )

for all x,y ∈X where a,b,c,e≥0 and a+b+c+e<1. Assume that pairs (TU,Q), (RS,P) are weakly compatible, and the pairs

(T,U), (T, Q), (R, S), (R, P) and (S, P) are commuting pairs of maps. Then T, U, R, S, Q and P have a unique common

fixed point in X.

Proof:Let x0∈X. by (2.1) we can define inductively a sequence {yn} in X such that y =RSx =Qx and

TUx =Px = y for all n=0,1, 2, 3 …

By(2.2), wehave

d(y ,y ) =d(RSx ,TUx )

≾

⎝ ⎜ ⎜ ⎜ ⎜ ⎜

⎛ ad(Px , Qx

)[d(Px , RSx ) + d(TUx , Qx )]

d(RSx , TUx ) + d(Qx , Px )

+bd(TUx , RSx )[d(TUx , Qx ) + d(Px , RSx )]

d(Px , Qx ) + d(TUx , RSx )

+cd(Qx , RSx ) [d(Px , TUx )]

d(RSx , TUx ) + d(RSx , Px )

+ed(Qx , RSx ) [d(TUx , Px )]

d(Px , Qx ) + d(TUx , RSx ) ⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎞

≾

⎝ ⎜ ⎜ ⎜ ⎜ ⎜

⎛ ad(y , y )

[d(y , y ) + d(y , y )]

d(y , y ) + d(y , y )

+bd(y , y )[d(y , y ) + d(y , y )]

d(y , y ) + d(y , y )

+cd(y , y ) [d(y , y )]

d(y , y ) + d(y , y )

+ed(y , y ). [d(y , y )]

d(y , y ) + d(y , y ) ⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎞

d(y ,y ) ≾ad(y , y )+bd(y ,y )

(1−b)d(y ,y )≾ad(y , y )

Therefore,

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Similarly we have,

d(y ,y )= d(TUx ,RSx )

= d(RSx , TUx )

≾

⎝ ⎜ ⎜ ⎜ ⎜ ⎜

⎛ ad(Px , Qx )

[d(Px , RSx ) + d(TUx , Qx )]

d(RSx , TUx ) + d(Qx , Px )

+bd(TUx , RSx )[d(TUx , Qx ) + d(Px , RSx )]

d(Px , Qx ) + d(TUx , RSx )

+cd(Qx , RSx ) [d(Px , TUx )]

d(RSx , TUx ) + d(RSx , Px )

+ed(Qx , RSx ) [d(TUx , Px )]

d(Px , Qx ) + d(TUx , RSx ) ⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎞

≾

⎝ ⎜ ⎜ ⎜ ⎜ ⎜

⎛ ad(y , y )

[d(y , y ) + d(y , y )]

d(y , y ) + d(y , y )

+bd(y , y )[d(y , y ) + d(y , y )]

d(y , y ) + d(y , y )

+cd(y , y ) [d(y , y )]

d(y , y ) + d(y , y )

+ed(y , y ). [d(y , y )]

d(y , y ) + d(y , y ) ⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎞

d(y , y )≾ad(y , y ) + bd(y , y )

Therefore,

(2.4)d(y , y )≾ a

1−bd(y , y )

(2.5) d(y , y )≾ k d(y , y ), where k=

Using (2.3), (2.4) and (2.5), we have

d(y , y )≾ k d(y , y )

≾ k2 d(y , y )

.

≾ kn d(y , y ) for all n.

(2.6) d(y , y )≾ knd(y , y ) for all n.

therefore,

d(yn,ym)≾ d(yn,yn+1)+ d(yn+1,yn+2)+… +d(ym-1,ym)

≾ (kn+kn+1+…+km-1)d(y1,y0)

≾ d(y1,y0), (using 2.6)

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Which implies that,d(y , y )→0as(n, m→ ∞). Hence {y } is a Cauchy sequence, by the completeness of X, there

exist z∈X such that,

(2.7) lim → RSx =lim → Qx =lim → TUx =lim → Px =z

Since, TU(X)⊆P(X) there exist u∈Xsuchthatz = Pu,

We claim that RSu=z, if possible RSu≠z, then by using (2.2), we have

d(RSu,z)=d(RSu,TUx )+d(TUx , z)

≾

⎝ ⎜ ⎜ ⎜ ⎛

ad(Pu, Qx )[ ( , ) ( , )]

( , ) ( , )

+bd(TUx , RSu)[ ( , ) ( , )]

( , ) ( , )

+cd(Qx , RSu) ₍ [ ( , )]

, ) ( , )

+ed(Qx , RSu) ₍ [ ( , )]

, ) ( , ) ⎠

⎟ ⎟ ⎟ ⎞

+d(TUx , z)

d(RSu,z)≾

ad(z, z)[ ( , ) ( , )]

( , ) ( , ) + bd(z, RSu)

[ ( , ) ( , )] ( , ) ( , ) +

cd(z, RSu) [ ( , )]

( , ) ( , )+ ed(z, RSu)

[ ( , )] ( , ) ( , )

+d(z, z)[using (2.7)]

taking the limit as n→ ∞, wehave

d(RSu,z)≾bd(z, RSu), which is a contradiction.Therefore,

(2.8) RSu=Pu=z

Since, RS(X)⊆Q(X) there exist v∈Xsuchthatz = Qv.

We claim that TUv=z, if possible TUv ≠ z, then by (2.2) and (2.8), we have

d(z,TUv)=d(RSu,TUv)

≾

⎝ ⎜

⎛ad(Pu, Qv)

[d(Pu, RSu) + d(TUv, Qv)]

d(RSu, TUv) + d(Qv, Pu) + bd(TUv, RSu)

[d(TUv, Qv) + d(Pu, RSu)] d(Pu, Qv) + d(TUv, RSu)

+cd(Qv, RSu) [d(Pu, TUv)]

d(RSu, TUv) + d(RSu, Pu)+ ed(Qv, RSu)

[d(TUv, Pu)]

d(Pu, Qv) + d(TUv, RSu)_⎠

⎟ ⎞

≾

ad(z, z)[ ( , )_{( ,} ₎( _{( , )}, )]+ bd(TUv, z)[ (_{( , )}, )₍ ( , )] , )

+cd(z, z) _{( ,}[ ( ,₎ )]_{( , )}+ ed(z, z) _{( , )}[ ( ₍, )] , )

[using (2.8)]

d(z,TUv) ≾ bd(TUv, z)

d(z,TUv) ≾bd(TUv, z),which is a contradiction.

therefore,TUv=Qv=z so,

(2.9) Pu=TUv=Qv=z

Similarly, Q and TU are weakly compatible maps, we have TUz=Qz.

Now we claim that z is a fixed point ofTU. If TUz≠z, then by (2.2), we have

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≾

⎝ ⎜

⎛ad(Pu, Qz)

[d(Pu, RSu) + d(TUz, Qz)]

d(RSu, TUz) + d(Qz, Pu) + bd(TUz, RSu)

[d(TUz, Qz) + d(Pu, RSu)] d(Pu, Qz) + d(TUz, RSu)

+cd(Qz, RSu) [d(Pu, TUz)]

d(RSu, TUz) + d(RSu, Pu)+ ed(Qz, RSu)

[d(TUz, Pu)]

d(Pu, Qz) + d(TUz, RSu)_⎠

⎟ ⎞

d(z,TUz) ≾

ad(z, TUz)[ ( , )_{( ,} ₎( , )]

( , ) + bd(TUz, z)

[ ( , ) ( , )] ( , ) ( , ) .

cd(TUz, z) [ ( , )]

( , ) ( , )+ ed(TUz, z)

[ ( , )] ( , ) ( , )

[using (2.8)]

d(z, TUz)≾cd(TUz, z) +e

2d(TUz, z)

d(z, TUz)≾(c + )d(TUz, z), which is a contradiction.

Therefore,TUz=z, hence Qz=z. So, we have

(2.10) TUz=Qz=z.

Similarly, P and RS are weakly compatible maps, we have RSz=Pz.

Now we claim that z is a fixed point of RS. If RSz ≠ , then by (2.2), we have

d(RSz, z)= d(RSz,TUz)

≾

⎝ ⎜

⎛ad(Pz, Qz)

[d(Pz, RSz) + d(TUz, Qz)]

d(RSz, TUz) + d(Qz, Pz) + bd(TUz, RSz)

[d(TUz, Qz) + d(Pz, RSz)] d(Pz, Qz) + d(TUz, RSz)

+cd(Qz, RSz) [d(Pz, TUz)]

d(RSz, TUz) + d(RSz, Pz)+ ed(Qz, RSz)

[d(TUz, Pz)] d(Pz, Qz) + d(TUz, RSz)_⎠

⎟ ⎞

we have,

d(RSz, z) ≾

ad(RSz, z)[ (₍ , ) ( , )]

, ) ( , ) + bd(RSz, z)

[ ( , ) ( , )] ( , ) ( , )

+cd(z, RSz) ₍ [ ( , )]

, ) ( , )+ ed(z, RSz)

[ ( , )] ( , ) ( , )

[using (2.10)]

d(RSz, z)≾ cd(z, RSz) +e

2d(z, RSz)

d(RSz, z)≾(c + )d(z, RSz), which is a contradiction.

Therefore, RSz=z. So, we have

(2.11) RSz=Pz=z.

So, z is a common fixed point of TU, Q, P and RS.

By commuting condition of pairs

Tz=T(TUz)=T(UTz)=TU(Tz)

Tz=T(Pz)=P(Tz) and Uz=U(TUz)=(UT)(Uz)=(TU)(Uz)

Uz=U(Pz)=P(Uz) which follows that, Tz and Uz are common fixed points of TU and P, Then

(2.12) Tz=z=Uz=Pz=TUz

Similarly, By commuting property,

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Rz=R(Qz)=Q(Rz) and Sz=S(RSz)=(SR)(Sz)=(RS)(Sz)

(Sz)=S(Qz)=Q(Sz) which follows that, Rzand Sz are common fixed points of RS and Q, Then

(2.13) Rz=z=Sz=Qz=RSz

Therefore z is a common fixed point of T, U, R, S, P and Q.

Uniqueness

Let w be other common fixed point of T,U,R,P,S and Q. if possible w ≠ .Then by (2.2), we have

d(z,w)=d(RSz,TUw)

≾

⎝ ⎜

⎛ad(Pz, Qw)

[d(Pz, RSz) + d(TUw, Qw)]

d(RSz, TUw) + d(Qw, Pz) + bd(TUw, RSz)

[d(TUw, Qw) + d(Pz, RSz)] d(Pz, Qw) + d(TUw, RSz)

+cd(Qw, RSz) [d(Pz, TUw)]

d(RSz, TUw) + d(RSz, Pz)+ ed(Qw, RSz)

[d(TUw, Pz)]

d(Pz, Qw) + d(TUw, RSz) _⎠

⎟ ⎞

=

⎝ ⎜

⎛ad(z, w)

[d(z, z) + d(w, w)]

d(z, w) + d(w, z) + bd(w, z)

[d(w, w) + d(z, z)] d(z, w) + d(w, z)

+cd(w, z) [d(z, w)]

d(z, w) + d(z, z)+ ed(w, z)

[d(w, z)] d(z, w) + d(w, z) _⎠

⎟ ⎞

= c +e

2 d(w, z)

d(z,w)≾ c + d(w, z),a contradiction.

So,z=w.

Hence, T, U, R, S, Q and P have a unique common fixed point in X.

REFERENCES

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2. Al Pervo: On the Cauchy problem for a system of ordinary differential equations. Pvi-blizhen met Reshen Diff Uvavn. Vol. 2, pp. 115-134, 1964.

3. C. Semple, M. Steel: Phylogenetics, Oxford Lecture Ser. In Math Appl, vol. 24, Oxford Univ. Press, Oxford, 2003.

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5. G. Junck: Commuting maps and fixed points. Am Math Monthly. vol. 83, pp. 261-263,1976.

6. L.G. Huang, X. Zhang: Cone metric spaces and fixed point theorem for contractive mappings. J Math Anal Appl. Vol. 332, pp. 1468-1476, 2007.

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9. S. Sessa, On a weak commutativity condition of mappings in fixed point consideration. PublInst Math, 32(46): 149-153(1982) 10. W. Chistyakov, Modular metric spaces, I: basic concepts. Nonlinear Anal. Vol. 72,pp. 1-14, 2010.

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BIOGRAPHY

Kamal Kumar is an assistant professor in Department of Mathematics at Pt. JLN Govt. P.G college, Faridabad, India. He has been teaching for 10 years. He has published 8 research papers in international journals. His research area includes fixed Point theory, analysis and Topology.

Nisha Sharma is an assistant professor in Department of Mathematics at Pt. JLN Govt. P.G college, Faridabad, India. She has been teaching for 2 years. She has published 10 research papers in international journals. Her research interest includes fixed Point theory, analysis and Topology.

Rajeev Jha is presently working in Department of mathematics, College of Engineering, TeerthankarUniverdsity, Maradabad, India. He has 19 years of experience in teaching and research. He has published 160 research papers in national/ International Journals and attended 50 national/International conferences in India and Abroad.

Ritu Sharma is an assistant professor in Department of Mathematics at Pt. JLN Govt. P.G college, Faridabad, India. She has been teaching for 5 years. Her research interest includes fixed Point theory and analysis.